Mathematics 244, section 1 – Linear Algebra Retest Exam 2 – March 28, 2007

Directions

Do all work in the blue exam booklet. There are 100 regular points and 5 extra credit points (see III C below).

I. All parts of this question refer to the matrix

A = ( \begin{matrix}1 & 4 & 2 & 1 & 2t \\ −3 & 2 & 2 & 3 & t^{2} \\ 2 & 1 & 0 & −1 & −t \end{matrix} )

A) (15) What is dim Nul(A) if t = 2? Find a basis for Nul(A) in this case. B) (10) What is dim Col(A) if t = 4. Find a basis for Col(A) in this case.

II. A) (15) Give the general proof of Cramer’s Rule for systems Ax = b with A ∈ M_{n×n} an

invertible matrix. B) (10) Give an example of a system Ax = b, where A is an invertible 3 × 3 matrix, but

the second component of the solution vector x is x_{2} = 0.

III. Let V = P_{2}(R), the vector space of polynomials of degree ≤ 2, and let W = M_{2×2}(R), the vector space of all 2 × 2 matrices with real entries.

A) (10) Show that T : V → W defined by T (p) = ( \begin{matrix}p(1) & p(2) \\ p′(1) & p′(2) \end{matrix} ) (p′ = derivative of p)

is a linear mapping.

B) (10) Let K = \{A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ W | dim Col(A) ≤ 1\} Is K a vector subspace of W?

Why or why not? C) (5 Extra Credit) Let T be the mapping from part A. What is Ker(T)? Justify your

answer.

IV. (True-False) For each true statement, give a short proof. For each false statement, give a counterexample. (Note: As always, “true” means “true in every case.” Partial credit will be given for definitions or theorems that apply even if you don’t find a complete solution.)

A) (10) If a, b, c are distinct real numbers, then the matrix V = ( \begin{matrix}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} ) is invertible.

B) (10) If A is a 5 × 7 matrix, then dim Nul(A) · dim Col(A) can equal 9. C) (10) Let

A = ( \begin{matrix}1 & −2 & 0 \\ −2 & −1 & −1 \end{matrix} )

Any 3 × 2 matrix X such that AX = 0_{2×2} has linearly dependent columns.